Chapter 7

Write each quotient in the form \(a+b i .\) See Example 5. $$ \frac{7}{4+3 i} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ x^{-2 / 5} \cdot x^{7 / 5} $$

Add or subtract. $$ -\frac{\sqrt[3]{2 x^{4}}}{9}+\sqrt[3]{\frac{250 x^{4}}{27}} $$

Simplify. See Examples 3 and 4 $$ \sqrt[4]{a^{8} b^{7}} $$

Simplify. Assume that the variables represent any real number. $$ \sqrt{(-7)^{2}} $$

Solve. \(\sqrt{2 x-1}-4=-\sqrt{x-4}\)

Write each quotient in the form \(a+b i .\) See Example 5. $$ \frac{9}{1-2 i} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ y^{4 / 3} \cdot y^{-1 / 3} $$

Rationalize each denominator. See Example 4. $$ \frac{-8}{\sqrt{y}+4} $$

Add or subtract. $$ \frac{\sqrt[3]{y^{5}}}{8}+\frac{5 y \sqrt[3]{y^{2}}}{4} $$

Simplify. See Examples 3 and 4 $$ \sqrt[5]{32 z^{12}} $$

Simplify. Assume that the variables represent any real number. $$ \sqrt[3]{(-8)^{3}} $$

Solve. \(\sqrt{2 x-1}=\sqrt{1-2 x}\)

Write each quotient in the form \(a+b i .\) See Example 5. $$ \frac{3+5 i}{1+i} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ 3^{1 / 4} \cdot 3^{3 / 8} $$

Rationalize each denominator. See Example 4. $$ \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}} $$

Simplify. Assume that the variables represent any real number. $$ \sqrt[5]{(-7)^{5}} $$

Solve. \(\sqrt{7 x-4}=\sqrt{4-7 x}\)

Use the properties of exponents to simplify each expression. Write with positive exponents. See Example 4 . $$ 5^{1 / 2} \cdot 5^{1 / 6} $$

Write each quotient in the form \(a+b i .\) See Example 5. $$ \frac{6+2 i}{4-3 i} $$

Simplify. See Examples 3 and 4 $$ \sqrt[3]{y^{5}} $$

Rationalize each denominator. See Example 4. $$ \frac{\sqrt{3}+\sqrt{4}}{\sqrt{2}-\sqrt{3}} $$

Simplify. Assume that the variables represent any real number. $$ \sqrt{4 x^{2}} $$

Solve. \(\sqrt{3 x+4}-1=\sqrt{2 x+1}\)

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{y^{1 / 3}}{y^{1 / 6}} $$

Write each quotient in the form \(a+b i .\) See Example 5. $$ \frac{5-i}{3-2 i} $$

Multiply and then simplify if possible. $$ \sqrt{7}(\sqrt{5}+\sqrt{3}) $$

Simplify. See Examples 3 and 4 $$ \sqrt{25 a^{2} b^{3}} $$

Simplify. Assume that the variables represent any real number. $$ \sqrt[4]{16 x^{4}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{x^{3 / 4}}{x^{1 / 8}} $$

Solve. \(\sqrt{x-2}+3=\sqrt{4 x+1}\)

Write each quotient in the form \(a+b i .\) See Example 5. $$ \frac{6-i}{2+i} $$

Multiply and then simplify if possible. $$ \sqrt{5}(\sqrt{15}-\sqrt{35}) $$

Rationalize each denominator. See Example 4. $$ \frac{2 \sqrt{a}-3}{2 \sqrt{a}+\sqrt{b}} $$

Simplify. Assume that the variables represent any real number. $$ \sqrt[3]{x^{3}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \left(4 u^{2}\right)^{3 / 2} $$

Solve. \(\sqrt{y+3}-\sqrt{y-3}=1\)

$$ (7 i)(-9 i) $$

Rationalize each denominator. See Example 4. $$ \frac{8}{1+\sqrt{10}} $$

Multiply and then simplify if possible. $$ (\sqrt{5}-\sqrt{2})^{2} $$

Simplify. Assume that the variables represent any real number. $$ \sqrt[5]{x^{5}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \left(32^{1 / 5} x^{2 / 3}\right)^{3} $$

Solve. \(\sqrt{x+1}-\sqrt{x-1}=2\)

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (-6 i)(-4 i) $$

Rationalize each denominator. See Example 4. $$ \frac{-3}{\sqrt{6}-2} $$

Multiply and then simplify if possible. $$ (3 x-\sqrt{2})(3 x-\sqrt{2}) $$

Simplify. Assume that the variables represent any real number. $$ \sqrt{(x-5)^{2}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{b^{1 / 2} b^{3 / 4}}{-b^{1 / 4}} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$$ (6-3 i)-(4-2 i) $$

Rationalize each denominator. See Example 4. $$ \frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}} $$